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The Donnelly-Fefferman theorem on $$q$$-pseudoconvex domains. (English) Zbl 1214.32015
The authors extend the notion of a $$q$$-subharmonic function introduced in the $$C^{2}$$ class by L.-H. Ho [Math. Ann. 290, No. 1, 3–18 (1991; Zbl 0714.32006)] to the class of upper semicontinuous functions, and use these functions to obtain $$L^{2}$$-estimations of Donnelly-Fefferman type for the $$\overline{\partial}$$-equations on certain domains with non-smooth boundaries. More precisely, let $$\varphi$$ be an upper semicontinuous function on an open set in $$\mathbb{C}^{n}$$; it is called $$q$$-subharmonic in $$U$$ if, for every linear complex space $$H$$ of dimension $$q$$, and for every compact set $$K\subset U\cap H$$, it has the following property: if $$h$$ is a continuous harmonic function on $$K$$ and $$h\leq\varphi$$ on $$\partial K$$, then $$h\leq\varphi$$ on $$K$$. Plurisubharmonicity is equivalent to 1-subharmonicity; an open set $$D$$ on $$\mathbb{C}^{n}$$ is called $$q$$-pseudoconvex if there is a $$q$$-subharmonic exhaustion function for $$D$$.
The main result is the following: Let $$D$$ be a $$q$$-pseudoconvex domain in $$\mathbb{C}^{n}$$, let $$\varphi$$ be a given $$q$$-subharmonic function in $$D$$, and let $$\psi\in C^{2}(D)$$ be strictly plurisubharmonic such that $$-e^{-\psi}$$ is $$q$$-subharmonic. Further, let $$\varepsilon\in(0,1)$$. Then for every $$\overline{\partial}$$-closed $$(0,r)$$-form $$g$$, with $$q\leq r\leq n$$, there is a solution of $$\overline{\partial}u=g$$ such that
$\int_{D}|u|^{2}e^{-\varphi+\varepsilon\psi}dV\leq\frac{4}{\varepsilon(1-\varepsilon)^{2}}\frac{1}{r}\mathop{\sum{'}}\limits_{| K|=r-1}\mathop{\sum}\limits_{j,k}\int_{D}\psi^{j\overline{k}}g_{jK}\overline{g}_{kK}e^{-\varphi+\varepsilon\psi}dV\tag{*}$
whenever the right hand side of $$(*)$$ is bounded. Here, $$(\psi^{j\overline{k}})=(\psi_{\nu\overline{\mu}})^{-1}$$.
As a corollary, the authors prove the following approximation result: Let $$D$$ be as above, $$h$$ a continuous $$q$$-subharmonic function in $$D$$, and $$K=\left\{ z\in D: h(z)\leq0\right\}\subset\subset D$$. If an $$\overline{\partial}$$-closed $$(0,r)$$-form $$f$$, $$r\geq q-1$$, is smooth in a neighborhood of $$K$$, then, for each $$\delta>0$$, there is a $$\overline{\partial}$$-closed $$(0,r)$$-form $$g_{\delta}$$ with coefficients in $$L^{2}(D)$$ such that $$\| f-g_{\delta}\|_{L^{2}(K)}<\delta$$.
The proof of the main result is done in two steps: first, the authors treat the case of a smoothly bounded $$q$$-pseudoconvex domain $$\Omega$$ and $$\varphi$$, $$\psi$$ smooth up to $$\overline{\Omega}$$, $$\psi$$ positive definite and $$-e^{-\psi}$$ $$q$$-subharmonic using the Bochner-Kodaira identity. The general case is approached by approximating $$D$$ by an increasing sequence $$\left\{ D_{\nu}\right\}$$, $$D_{\nu}\subset\subset D$$, and choosing a decreasing sequence $$\left\{ \varphi_{\mu}\right\}$$ of smooth $$q$$-subharmonic functions which converges pointwise to $$\varphi$$, and applying the estimate obtained before, with $$\varphi_{\nu}$$, $$w=e^{-\varepsilon\psi}$$ and $$\Omega=D_{\nu}$$.

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32F10 $$q$$-convexity, $$q$$-concavity
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##### References:
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